Based on what has happened in the past and what he has said recently, he would rather end the marriage than do more with open marriage.

I don't know how I can go back to monogamy--I'm very afraid I will cheat again.

...we had a conversation where I basically decided that there were no lines for me sexually, and that I couldn't promise him that I wasn't going to cross his lines. He decided he couldn't live with me having a polyamorous lifestyle, and separated from me. A week later I asked him if he would take me back if I said I would not live the lifestyle, and if he could forgive the cheating. He said yes.

The one thing he said he might be open to is exploring a relationship with a third who is a woman.

Hi there!

I boldfaced my red flags, since Redpepper already pointed out the "we all live together, with kids!" thing.

So let me re-capture if I got this right, and please correct me if I haven't understood you correctly.

1. He wants a divorce if you continue having extra-marital relationships.
2. You don't want to continue in a monogamous marriage.
3. You two cannot come up with boundaries/lines you both could respect.
4. You promised, against your better knowledge, under the threat of divorce, that you will stick to his lines. You lied, and he chose to believe you.
5. Now you find yourself in a situation where you try to solve this impasse by imagining a fantasy relationship everyone could be okay with, and when you find it doesn't suit reality, that he is still unhappy, you ask how you could manipulate him into changing his mind and feeling better.

Frankly, there's no ethical way to make someone abandon their belief system and to change their emotional make-up IF THEY DON'T WANT TO, and your husband has clearly and repeatedly indicated that he does not.

I understand you have small children, but you have also fundamentally incombatible wants for your life together. No amount of biblical argumentation can change that.